Question regarding frequency resolution

[Arduryo]

Hello, I have a question regarding the octavefilter function. I have a signal representing sound pressure in Pascals over time t, from which I wish to calculate the third-octave spectra. When calculating manually, a conversion to the frequency domain using FFT is necessary; so far, I have used FFTs with 2^15 samples, a Von Hann window and a 50% overlap. This corresponds to a frequency resolution of approximately 6 Hz. What is the frequency resolution in the result of the octavefilter function? So far, I have been using the octavefilter function in the range from 12 Hz to 20,000 Hz (i.e. the default setting), with fraction=3 and a sampling rate of 100,000 Hz. Perhaps I didn’t find this in the Readme either, but how does it work exactly?

Greetings

[jmrplens]

Hi, thanks for the question.

octavefilter does not use an FFT/Welch-style spectrum internally, so it does not have a frequency resolution in the sense of fs / nfft.

It is a time-domain fractional-octave filter bank. For fraction=3, the output contains one level per third-octave band. The relevant frequency granularity is therefore the standardized band spacing and the lower/upper band edges, not FFT bin spacing.

The figure below uses a synthetic signal only to illustrate the distinction:

For your parameters (fs=100000, limits=[12, 20000], fraction=3), PyOctaveBand returns 33 third-octave bands. Some examples are:

Nominal band	Lower edge	Center	Upper edge	Bandwidth
12.5 Hz	11.22 Hz	12.59 Hz	14.13 Hz	2.91 Hz
100 Hz	89.13 Hz	100.00 Hz	112.20 Hz	23.08 Hz
1 kHz	891.25 Hz	1000.00 Hz	1122.02 Hz	230.77 Hz
10 kHz	8912.51 Hz	10000.00 Hz	11220.18 Hz	2307.68 Hz
20 kHz	17782.79 Hz	19952.62 Hz	22387.21 Hz	4604.42 Hz

The bandwidth increases with frequency because octave and fractional-octave bands are logarithmically spaced. In other words, the relative bandwidth is approximately constant, while the absolute bandwidth in Hz grows with frequency.

Internally, octavefilter computes the standardized band center frequencies and edges, designs a band-pass IIR filter for each band, and applies those filters in the time domain using second-order sections. For low-frequency bands it may internally downsample before filtering to improve numerical stability. It then calculates the RMS or peak level of each filtered band.

So if you need narrow FFT bins for tonal inspection, a Welch/FFT workflow is the right tool. If you need standardized third-octave band levels, octavefilter directly estimates the energy in those standardized bands.

You can inspect the exact bands for your setup with:

from pyoctaveband import getansifrequencies

fc, fl, fu, labels = getansifrequencies(fraction=3, limits=[12, 20000])
for label, center, lower, upper in zip(labels, fc, fl, fu):
    print(label, center, lower, upper, upper - lower)

If you also want to inspect which narrowband FFT/Welch bins fall inside one of the third-octave bands, run Welch on the original pressure signal and use the PyOctaveBand band edges as masks:

import numpy as np
from scipy import signal
from pyoctaveband import octavefilter, getansifrequencies

fs = 100_000
x = pressure_signal_pa  # 1D pressure signal in Pa

# Standardized third-octave levels from PyOctaveBand.
levels, centers = octavefilter(
    x,
    fs=fs,
    fraction=3,
    limits=[12, 20_000],
    detrend=True,
)

# Same standardized band definitions, including lower/upper edges.
fc, fl, fu, labels = getansifrequencies(fraction=3, limits=[12, 20_000])

# Narrowband Welch estimate on the original signal.
nperseg = min(2**15, len(x))
freq_bins, psd = signal.welch(
    x,
    fs=fs,
    window="hann",
    nperseg=nperseg,
    noverlap=nperseg // 2,
    scaling="density",
)

# Example: list the Welch bins inside the third-octave band closest to 1 kHz.
band_index = int(np.argmin(np.abs(np.asarray(fc) - 1000.0)))
in_band = (freq_bins >= fl[band_index]) & (freq_bins <= fu[band_index])

print("Selected third-octave band:", labels[band_index], fl[band_index], fc[band_index], fu[band_index])
print("Welch bin spacing:", freq_bins[1] - freq_bins[0], "Hz")
print("Welch bins inside this band:")
for f, pxx in zip(freq_bins[in_band], psd[in_band]):
    print(f, pxx)

That keeps the two concepts separate: PyOctaveBand gives standardized fractional-octave band levels, while Welch gives narrowband FFT bins.

One small note: with fs=100000 and 2^15 FFT samples, the bin spacing is about 3.05 Hz (fs / nfft). A value around 6 Hz would correspond to 2^14 samples at 100 kHz, or to a different sampling rate/FFT length. The 50% overlap improves averaging, but it does not change the frequency-bin spacing of each FFT segment.

I agree this distinction could be clearer in the README, so I will add a short note explaining that octavefilter is a time-domain filter-bank method rather than an FFT-bin method.